Section 5.2 Properties of Hermitian Matrices
The eigenvalues and eigenvectors of Hermitian matrices Section 5.1 have some special properties.
The eigenvalues of a Hermitian matrix must be real.
To see why this relationship holds, start with the eigenvector equation
and multiply on the left by \(\langle v|\) (that is, by \(v^\dagger\)):
But we can also compute the Hermitian conjugate (that is, the conjugate transpose) of (5.2.1), which is
Using the fact that \(M^\dagger=M\text{,}\) and multiplying by \(|v\rangle\) on the right now yields
Comparing (5.2.2) with (5.2.4) now shows that
as claimed.
Eigenvectors corresponding to different eigenvalues must be orthogonal.
The argument establishing this relationship is similar to the one above. Suppose that
Then
or equivalently
Thus, if \(\lambda\ne\mu\text{,}\) \(|v\rangle\) must be orthogonal to \(|w\rangle\text{.}\)
In the case of repeated eigenvalues, it is possible to make a basis orthogonal using a process called Gram-Schmidt orthogonalization. (This amounts to simply subtracting off the parts of a given basis that are not orthogonal.)
Conclusion:.
It is always possible to find an orthonormal basis of eigenvectors for any Hermitian matrix.
Sensemaking 5.1. Properties of Anti-Hermitian Matrices.
Repeat the two proofs above to find similar properties for anti-Hermitian matrices.